Special metrics in supergravity and new G-structures

String theories have been a nice source of connections between physics and geometry. The supersymmetry equations impose topological and differential conditions on the space-time manifold, which can be codified in terms of G-structures. This project proposes a systematic investigation of solutions in 11-dimensional supergravity theory based on special metrics (e.g. Einstein metrics) and G-structures. Our goal is to facilitate the machinery of geometries with torsion and discover new relations with M-theory, by developing new supergravity backgrounds. We also suggest a detailed description of the special geometry related to the quaternionic real form of the complex special orthogonal Lie group and interactions with parabolic/spin geometries. Some emphasis will be given to the construction of Einstein metrics on principal torus bundles over homogeneous spaces. The project will have potential applications in M-theory, will deepen our knowledge of homogeneous geometric structures and quaternion geometries and we will establish new operators of Dirac type and new Einstein metrics.

The first part of our state of art seeks to contribute to a better understanding of the geometry of solutions in eleven-dimensional supergravity and the relation with special metrics, especially Einstein metrics. We attempt to construct new bosonic supergravity backgrounds in eleven dimensions in terms of special geometric structures and suitable Lorentzian geometries. A further aim is to answer the existence problem of super-symmetries, which represent the fermionic part of the field equations. This should be treated in terms of spin geometry.
In the second part we intend to present two new types of G-structures based on the Lie groups SO*(2n) and Sp(1)SO*(2n), respectively. The reduction of the structure group of a 4n-dimensional smooth manifold to these Lie groups induces new kinds of geometric structures, where symplectic and (almost) hypercomplex or (almost) quaternion geometries, respectively, become compatible in a delicate way. Differential invariants adapted connections and intrinsic torsion are basic notions appearing in this direction. The interactions with parabolic geometries and the theory of ODEs, are also included in the list of research goals.
The last part is devoted to the construction and classification of new invariant Einstein metrics on compact homogeneous spaces, in particular on non-Kählerian C-spaces. Although in the last years there is remarkable progress towards the classification of homogenous Einstein metrics on flag manifolds and other kinds of compact homogeneous spaces, we still know very few of the classification of invariant Einstein metrics on compact simply connected homogeneous complex manifolds which are not Kähler. Such manifolds are usually even dimensional torus bundles over flag manifolds. In this project, we aim to provide the very first classification results for several homogeneous spaces of this kind. Parallel, we aim to provide Ricci flow techniques suitable for the study of homogeneous Einstein metrics. Such an analytical approach is based on the rich theory of ODEs and more generally of dynamical systems. Finally, the special almost Hermitian geometry of C-spaces or the existence of other types of suitable reductions, e.g. Spin(7)-structures or PSU(3)-structures in dimension eight, are other remarkable related research tasks.